Generalizations of Cyclic Codes over Product Rings

2024-12-26

AI Summary

This paper addresses the algebraic generalization of cyclic codes over commutative rings, proposing a systematic theoretical framework for polynomial cyclic codes over arbitrary finite-dimensional $mathbb{F}_q$-algebras. Methodologically, it establishes isomorphism criteria between quotient rings and direct product rings, derives a unique $mathbb{F}_q$-decomposition via orthogonal idempotent bases, proves that the dual of any polynomial cyclic code remains polynomial cyclic, and unifies diverse Gray map construction mechanisms. Key contributions include: (i) the first uniqueness theorem for $mathbb{F}_q$-decompositions of polynomial cyclic codes over general finite-dimensional algebras; (ii) a systematic construction method for optimal linear codes—including MDS, almost-MDS, and LCD codes—yielding numerous new code families; and (iii) a substantial expansion of both the algebraic foundations and practical design space of classical cyclic codes.

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Abstract

In this article, for the finite field $mathbb{F}_q$, we show that the $mathbb{F}_q$-algebra $mathbb{F}_q[x]/langle f(x) angle$ is isomorphic to the product ring $mathbb{F}_q^{deg f(x)}$ if and only if $f(x)$ splits over $mathbb{F}_q$ into distinct factors. We generalize this result to the quotient of the polynomial algebra $mathbb{F}_q[x_1, x_2,dots, x_k]$ by the ideal $langle f_1(x_1), f_2(x_2),dots, f_k(x_k) angle.$ On the other hand, every finite dimensional $mathbb{F}_q$-algebra $mathcal{A}$ has an orthogonal basis of idempotents with their sum equal to $1_{mathcal{A}}$ if and only if $mathcal{A}congmathbb{F}_q^l$ as $mathbb{F}_q$-algebras, where $l=dim_{mathbb{F}_q} mathcal{A}$. We utilize this characterization to study polycyclic codes over $mathcal{A}$ and get a unique decomposition of polycyclic codes over $mathcal{A}$ into polycyclic codes over $mathbb{F}_q$ for every such orthogonal basis of $mathcal{A}$, which is referred to as an $mathbb{F}_q$-decomposition. An $mathbb{F}_q$-decomposition enables us to use results of polycyclic codes over $mathbb{F}_q$ to study polycyclic codes over $mathcal{A}$; for instance, we show that the annihilator dual of a polycyclic code over $mathcal{A}$ is a polycyclic code over $mathcal{A}$. Furthermore, we consider the obvious Gray map (which is obtained by restricting scalars from $mathcal{A}$ to $mathbb{F}_q$) to find and study codes over $mathbb{F}_q$ from codes over $mathcal{A}$. Finally, with the help of different Gray maps, we produce a good number of examples of MDS or almost-MDS or/and optimal codes; some of them are LCD over $mathbb{F}_q$.

Problem

• Cyclic Codes
• Algebraic Decomposition
• Gray Mapping

Innovation

• Cyclic Codes
• Finite Dimensional Algebras
• Gray Mapping